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Mirrors > Home > ILE Home > Th. List > ifcldadc | Unicode version |
Description: Conditional closure. (Contributed by Jim Kingdon, 11-Jan-2022.) |
Ref | Expression |
---|---|
ifcldadc.1 | |
ifcldadc.2 | |
ifcldadc.dc | DECID |
Ref | Expression |
---|---|
ifcldadc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iftrue 3449 | . . . 4 | |
2 | 1 | adantl 275 | . . 3 |
3 | ifcldadc.1 | . . 3 | |
4 | 2, 3 | eqeltrd 2194 | . 2 |
5 | iffalse 3452 | . . . 4 | |
6 | 5 | adantl 275 | . . 3 |
7 | ifcldadc.2 | . . 3 | |
8 | 6, 7 | eqeltrd 2194 | . 2 |
9 | ifcldadc.dc | . . 3 DECID | |
10 | exmiddc 806 | . . 3 DECID | |
11 | 9, 10 | syl 14 | . 2 |
12 | 4, 8, 11 | mpjaodan 772 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 682 DECID wdc 804 wceq 1316 wcel 1465 cif 3444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-if 3445 |
This theorem is referenced by: updjudhf 6932 omp1eomlem 6947 difinfsnlem 6952 ctmlemr 6961 ctssdclemn0 6963 ctssdc 6966 enumctlemm 6967 xaddf 9582 xaddval 9583 iseqf1olemqcl 10214 iseqf1olemnab 10216 iseqf1olemjpcl 10223 iseqf1olemqpcl 10224 seq3f1oleml 10231 seq3f1o 10232 exp3val 10250 xrmaxiflemcl 10969 summodclem2a 11105 zsumdc 11108 fsum3 11111 isumss 11115 fsum3cvg2 11118 fsum3ser 11121 fsumcl2lem 11122 fsumadd 11130 sumsnf 11133 sumsplitdc 11156 fsummulc2 11172 isumlessdc 11220 cvgratz 11256 eucalgval2 11646 lcmval 11656 ennnfonelemg 11827 subctctexmid 13092 |
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